How is this an exponential curve?

Question

cgo 2015-1-14

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/169871-how-is-this-an-exponential-curve

编辑： Shoaibur Rahman 2015-1-14

采纳的回答： Shoaibur Rahman

在 MATLAB Online 中打开

Hello,

I would like your help on the following code.

How does this trace an exponential curve? N=8000; % number of steps to take T=8; % maximum time h=T/N; % time step t=(0:h:T); % t is the vector [0 1h 2h 3h ... Nh] y=zeros(size(t)); % prepare place to store locations

y(1)=3;             % initial height
for i=1:N           % start taking steps
  y(i+1)=y(i)-y(i)*h;
end;
plot(t,y), hold on  % plot more permanently
y(1)=-2;            % initial height
for i=1:N           % start taking steps
  y(i+1)=y(i)-y(i)*h;
end;
plot(t,y);          % plot more permanently

Please help.

Thanks

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Answer 1

Shoaibur Rahman 2015-1-14

1
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/169871-how-is-this-an-exponential-curve#answer_164824

编辑：Shoaibur Rahman 2015-1-14

在 MATLAB Online 中打开

Convert your difference equation into differential equation:

y(i+1) = y(i)-y(i)*h;
y(i+1) - y(i) = -y(i)*h;
(y(i+1) - y(i))/h = -y(i);
dy/dt = -y

Time step h is replaced by dt to make the equation more understandable. Now, solve this differential equation using variable separation method:

dy/y = -dt
ln|y| = -t + C
y = exp(-t+C)
y = A*exp(-t)

At t = 0, A = y(0) = 3, in your case, y(0) is equivalent to y(1) in Matlab

y = 3*exp(-t)

This means that y is exponential. However, when solving using difference equation, make sure that h or the time is small enough to keep the solution stable. For this particular equation, h<1 . Use a larger value of h in your code just to see the effect.